Differential Antisymmetric Infinitesimal Bialgebras, Coherent Derivations and Poisson Bialgebras
Yuanchang Lin, Xuguang Liu, Chengming Bai

TL;DR
This paper develops a new bialgebra theory for differential algebras, introducing differential antisymmetric infinitesimal (ASI) bialgebras, and explores their connections to Poisson bialgebras, derivations, and related algebraic structures.
Contribution
It generalizes ASI bialgebras to differential algebras, introduces coherent derivations, and extends Poisson bialgebra constructions within this framework.
Findings
Characterization of differential ASI bialgebras via double constructions and matched pairs.
Introduction of $ ext{O}$-operators and differential dendriform algebras for constructing these bialgebras.
Extension of Poisson algebra and bialgebra theory using differential Zinbiel algebras.
Abstract
We establish a bialgebra theory for differential algebras, called differential antisymmetric infinitesimal (ASI) bialgebras by generalizing the study of ASI bialgebras to the context of differential algebras, in which the derivations play an important role. They are characterized by double constructions of differential Frobenius algebras as well as matched pairs of differential algebras. Antisymmetric solutions of an analogue of associative Yang-Baxter equation in differential algebras provide differential ASI bialgebras, whereas in turn the notions of -operators of differential algebras and differential dendriform algebras are also introduced to produce the former. On the other hand, the notion of a coherent derivation on an ASI bialgebra is introduced as an equivalent structure of a differential ASI bialgebra. They include derivations on ASI bialgebras and the set of…
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Taxonomy
TopicsAdvanced Topics in Algebra · Algebraic structures and combinatorial models · Nonlinear Waves and Solitons
